The conjecture would be useful for motivic homotopy
theory, since \(\mathbf {G}_m\) -actions occur everywhere. When \(U\)  is smooth,
both \(Y\)  and \(U\)  are unions of affine bundles over the connected components
of \(Y^T\) , by Białynicki-Birula [1]}. But even then,
we only know how to prove that \(Y\rightarrow U\)  is an \(\mathbf {A}^1\) -homotopy equivalence
after a suitable suspension (Theorem REF ).
Regardless of whether \(U\)  is smooth,
the conjecture would be clear if the \(T\) -action on \(U\) 
extended to a morphism
\(\mathbf {A}^1\times U\rightarrow U,\) 
